Image statistics have played an important role in image processing. In particular, a variety of statistical models have been proposed in the background art for various applications such as image filtering, image coding, image restoration, and image analysis. Moreover, many image processing applications are even impossible to pursue without appropriate statistical models.
The distribution of the JPEG discrete cosine transform (DCT) coefficients can be modeled as generalized Laplacian distribution or generalized Cauchy distribution. However, the probability distribution of the most significant digit of the JPEG DCT coefficients has not been reported in the background art.
Benford's law, also known as the first digit law or significant digit law, is an empirical law which states that the probability distribution of the first digit, x (where x=1, 2, . . ., 9), in a set of natural numbers is logarithmic. The Benford's law was originally proposed by F. Benford in the paper entitled: “The law of anomalous numbers,” Proc. Amer. Phil. Soc., vol. 78, pp. 551-572, 1938. In particular, if a data set satisfies Benford's law, its significant digits will have the following distribution:p(x)=log10(1+(1/x)), where x=1,2, . . . , 9, and where p(x) stands for probability of x.  (1)
The validity of Benford's law has been demonstrated and verified in various domains. While the naturally generated data obey the Benford's law well, the maliciously altered data do not follow this law in general. This property has been widely used in the fraud detection and accounting areas.
Applications of Benford's law in image processing field have been explored by very few researchers and primarily in recent years. One background art example is given by J. M. Jolion, “Images and Benford's law,” Journal of Mathematical Imaging and Vision 14, 73-81, 2001. Jolion showed that the magnitude of the gradient of an image obeys this law and gives some possible applications in image processing such as entropy coding. Another example of the background art is given by E. Acebo, and M. Sbert, “Benford's law for natural and synthetic images,” Computational Aesthetics in Graphics, Visualization and Imaging, 2005. Acebo and Sbert demonstrated how light intensities in natural images, under certain constraints, obey the Benford's law closely. However, there are no known previous investigations of the relationship between the distributions of the block-DCT coefficients and Benford's law.
Yet another background art application is given in Z. Fan and R. L. Queiroz, “Identification of bitmap compression history: JPEG detection and quantizer estimation,” IEEE Transaction on Image Processing, vol. 12, no. 2, February 2003. Fan and Queiroz proposed a JPEG compression detection scheme based on the detection of the blockiness artifacts introduced by JPEG compression. A maximum likelihood estimation method is proposed in their paper to estimate the JPEG quantization table after a JPEG image has been detected. Although their approach demonstrates some good results, its performance at very high compression quality (Q-factor>90) is rather limited and it fails when Q-factor is larger than 95. Therefore, there is a need in the art for expanded application of Benford's law and improved performance for the law in very high compression factors image processing applications.